Its research found 44% of voters want Clarke, the justice secretary, to remain in the Cabinet, compared with 27% who want him out of office. That contrasts with 43% wanting Cameron to fire his chancellor, compared to 28% thinking Osborne should remain at No 11.

Clarke has attracted controversy during his two years in power via the 'catgate' scandal and his comments about rape, but his liberal stance on criminal justice issues has placated Liberal Democrats and won praise from pro-rehabilitation reformers.

Last week the 72-year-old insisted he was "totally laid back about a reshuffle", after newspaper reports suggested he was resisting the idea of losing his job halfway through the implementation of his liberalising reforms.

Right-wing Conservative MPs have been agitating for some months for Clarke to be replaced by a Tory minister with firmer right-wing attitudes about law and order, like former shadow home secretary Chris Grayling.

If Cameron chose to follow the whims of voters he would also be obliged to sack health secretary Andrew Lansley and Tory minister without portfolio Sayeeda Warsi.

Liberal Democrat David Laws could replace one of them, the poll suggests. He was forced to resign after less than a month in the Treasury as its chief secretary in 2010 over misleading expenses claims, but his return is now backed by 44% of voters.

Only 16% believe he should remain out of power. He quit after it emerged he had claimed expenses on a London flat to conceal the fact the landlord was his gay lover.